# (how) are vector bundles and homotopy groups related?

Hello,

homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the case of X = S^1 (the 1-sphere), the isomorphism classes of 1-bundles correspond to (the generators of) π___{1}(S^1), since there is the trivial bundle, the Moebius bundle, and that's it. So my question is: am I right with this, and if yes: what needs to happen that π_n(X) = {homotopy classes of maps:grassmannian -> X}?

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The map goes the other way: vector bundles over X correspond to homotopy classes of maps from X into a grassmannian.

Let BO(n) be the grassmannian of n-plane bundles in R-infinity. Then, if you want to know about n-dimensional real vector bundles over Sk you are led to study the homotopy classes of maps from Sk to BO(n), or in other words πk BO(n).

In particular, the fact that there are exactly two 1-bundles over S1 comes from a calculation π1 BO(1)=Z/2.

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Thank you (both) very much, now it makes sense! –  knot Oct 26 '09 at 20:55